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INSTRUCTION S

FOR

YOUNG STUDENTS.

I would advise the young Practitioner to draw every Figure as he proceeds; carefully remarking what things, as Points, Lines, Angles, &c. are given; which are, in general, ftronger marked than the operative Lines; they being either dotted or finer drawn; the given Lines, &c. are, by that means, obvious and diftinguishable from the other.

In Geometry, as in Arithmetic, there is always fome Data or things given; from which, in Theory, other Properties are deduced, as a neceffary confequence; and, in Practice, fomewhat is required to be done, or performed, from what is given.

Let the Practitioner, therefore, felect the given things, and mark them down, firft, in the pofition given in the Premifes; but, with as much variation as it will admit of; i, e. he need not put them exactly as in the figure, only obferve that they are as required.

e. g. In Prob. 4. an Angle is required to be made at the extremity of a given Line; but the pofition of that given Line is not abfolutely determined; alfo, the Angle may be made at either extreme, and either above or below the given Line.

Likewife, in the 6th and 7th Problems, the Perpendicular may be drawn, and the Point, in the 7th, given on either Side of the Line; for, let it be obferved, and carefully remembered, that, by the Term Perpendicular, nothing more is meant than the Pofition one Line has to another; which Pofition, is when they make a Right Angle or Right Angles with each other; no regard being had to the pofition or fituation of either, separately.

These things being premifed, and the given Lines, &c. de fcribed on Paper; carefully obferve the directions given in the operation, and proceed accordingly, ftep by step, drawing every Line, Angle, Ark, &c. as the Problem directs.

PRAC

29

PRACTICAL

GEOMETRY.

PROBLEM I.

To defcribe a Circle of any given Radius, and on a given Center.

AB is the Radius given, and C the given Center.

FIX

IX one Point of a pair of Compaffes in either extreme of the given Line, AB,

and extend the other Point to the other ex- : treme, i. e. open the Compaffes equal to the given line.

Then, fix one Point of the Compaffes in C, the Center given, and revolve the other Point

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around; which, by its revolution, will defcribe the Circumference DEF.

DEF contains the Circle required (Def. 19.)

Poft. 4.

For, if a Right Line be drawn from the Center to the Circumference, as CD, it is equal to the given Line AB; by Conftruction.

PRO.

B

30

PRACTICAL GEOMETRY.

PROBLEM

ÍI. 2nd. I. Euclid.

To draw Right Line from a Point given, C, which fhall be equal to a given Line; AB.

Extend or open the Compaffes equal to the Line given; fix one of their Points in C, the given Point, and, with the other, defcribe a Circle, or portion of a Circle only, at D. (Pr. 1.)

Apply a ftreight Ruler close to the Point C, and croffing the Ark at D, (in whatever pofition you require the Line to be) and, with the point of a Pencil or a Drawing Pen (applied firft to the Point C, and drawn, along the edge of the Ruler, to the Ark at D) defcribe the Right Line CD. (Post.2.) Which is equal to the given Line AB; by Conftruction; and by N. B. Def. 20.

SCHOL. Thus, the genéfis of a Right Line (Def. 3.) is conceived to be by the direct motion of a Point.

B. In the practice of Geometry, it is often required to draw or to make a Line equal to another Line given ; which is done by drawing an Ark of a Circle, as AB, from the given Line, AC, till it cuts the other; if the two Lines, AC and CB, touch ať the Point given, C, which is made the Center.

But, if they do not touch, the Line given is taken for Radius (as in the Problem) and an Ark drawn where it is required; for equal Circles have equal Radii, as well as all Radii of the fame Circle are equal; which needs no other Demonftration than the genesis of a Circle, in N. B. Def. 20. So that, hereafter, when two Lines are obferved to be Radii of the fame Circle, it is fufficient Demonftration that the Lines are equal; and alfo, when they are made Radii of equal Circles.

APPL. The Application of this Problem, in defigning, is to delineate or draw, on Paper, &c. a Right Line equal to fome known measure, as AB, by a Scale of equal Parts.

PRO

PROBLEM

III. 3. I. Euclid.

A Right Line being given, to cut off a por ion or fegment equal to another given Line, or known measure.

AB is the first given Line, and CD the measureofthe Segment required to be cut off.

Extend the Compaffes from C to D, i. e. with the Radius CD, fetting one Point of the Compaffes in either extreme of AB, (from which the given Segment is required to be cut off), as A, with the other Point, draw a small Ark cutting the Line AB, at E. QE. F.

2. After the fame manner, any portion of an Ark of a Circle may be cut off. As AeB,

B

E

APPL. In delineating, the given Line A B may be fuppofed to be an indefinite Line already drawn; or, it may reprefent a certain meafure, by fome Scale, fuppofe 5 Feet, being made equal to 5 Divifions on the Scale; and it is required to cut off 3 Feet from the extreme Point A, of the Line AB, or two Feet from the other extreme, B.

By which means, a Right Line may be divided in any Proportion required.

FRO.

PROBLEM

IV.

23. I. Euclid.

B

D

To make an Angle, equal to a right-lined Plane
Angle given.

ABC is the given Angle, and DE a Line given ; it is required to make, at the point E, an Angle, with the Line DE, equal to the given Angle.

With any Radius, at difcretion, on the VerC tex of the given Angle, B, describe an Ark, a b, cutting the two Sides AB and BC in the Points a and b.

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Poft.

4.

With the fame Radius, on the Point E, draw the Ark DF; take the measure of the Ark ab

in your Compaffes; make DF equal ab.
Draw EF through the Point F; and it is done;

2. Pr. 3.

Q. E. F.

The Angle DEF is equal to the given Angle ABC; i. e. FE inclines the fame to ED, as AB to BC.

This is evident, from the Theory of Plane Angles, Art, 4. a b and DF being equal portions of equal Circles. Or, by drawing the Chord Lines ab and DF, the Triangles Bab, EDF, are Congruous. And the Angle DEF is equal ABC.

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Con.

P. 7. 1.

APPL. From this Problem we learn to delineate, i. e. to lay down or draw, on Paper, any right-lined Angle, of a piece of Ground or Building, &c. which we have measured, to form a Plan of it.

The

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